Theresa May speaks to workers from Ørsted, a Danish energy and wind turbine firm, in Grimsby.
Photograph: Darren Staples/EPA

Theresa May appears set for a second humiliating defeat when she brings her Brexit deal back to parliament next week, after the EU’s chief negotiator, Michel Barnier, rebuffed her pleas for last-minute concessions.

The prime minister urged MPs to “get it done” and back her deal, in an impassioned speech at a dockside warehouse in the leave-voting town of Grimsby.

A vote against the deal would mean “not completing Brexit and getting on with all the other important issues people care about, just yet more months and years arguing”, May told MPs. “If we go down that road we might never leave the EU at all.”

Addressing workers from Ørsted, a Danish energy and wind turbine firm, May also urged the EU to make new concessions over the Irish backstop – the issue that caused many of her MPs to vote against the deal the first time – before last-ditch talks in Brussels this weekend.

The EU “has to make a choice too”, the prime minister said. “We are both participants in this process. It is in the European interest for the UK to leave with a deal. We are working with them but the decisions that the European Union makes over the next few days will have a big impact on the outcome of the vote.

“European leaders tell me they worry that time is running out and that we only have one chance to get it right. My message to them is: now is the moment for us to act.”

But Barnier immediately appeared to rebuff the prime minister, by responding with an offer of reverting to his original plan, the Northern Ireland-only backstop, which May repeatedly said no prime minister could accept, because it risked creating a border in the Irish Sea.

The EU’s chief negotiator said in a series of tweets that the EU was committed “to give the UK the option to exit the single customs territory unilaterally, while the other elements of the backstop must be maintained to avoid a hard border. [The] UK will not be forced into a customs union against its will.”

The Brexit secretary, Stephen Barclay, immediately replied: “With a very real deadline looming, now is not the time to rerun old arguments. The UK has put forward clear new proposals. We now need to agree a balanced solution that can work for both sides.”

The Northern Ireland-only backstop was vehemently rejected by the government’s partners in the DUP, who fear that it would effectively sever Northern Ireland from the rest of the UK by requiring checks as goods pass back and forth.

And the DUP’s deputy leader, Nigel Dodds, said on Friday: “This is neither a realistic nor a sensible proposal from Michel Barnier. It disrespects the constitutional and economic integrity of the United Kingdom.”

After May lost the meaningful vote by a majority of 230 in January, she promised to seek “legally binding changes” to the backstop. But, while talks are expected to continue over the weekend, government insiders have become increasingly gloomy about the prospects for a last-minute shift.

One cabinet source said the mood was so bleak some ministers were speculating the defeat could even be almost as large on Tuesday. “Feedback suggests that it’s in a similar place to where it was last time the vote was held,” the source said.

May has pledged to hold two further votes next week if the vote is rejected on Tuesday – allowing MPs the opportunity to rule out no deal, and to delay Brexit.

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The Guardian understands another plan under discussion is to hold a third meaningful vote immediately after no deal has been ruled out – in the hope of focusing the minds of Brexiter rebels.

The government has pinned its hopes on negotiating changes that will allow the attorney general, Geoffrey Cox, to revise his legal advice warning the UK could be trapped indefinitely in the backstop.

In his response to the prime minister, Barnier reiterated the current powers of the arbitration panel to suspend elements of the backstop, should the EU be found to have breached its commitment to negotiate in “good faith” to find an alternative.

The British negotiators have been seeking to persuade the EU to go further and allow the UK to free itself from the backstop if the panel found it had made all reasonable efforts to find an alternative solution. Brussels has regarded that move as an attempt to slip in a unilateral exit mechanism to the withdrawal agreement.

Barnier said he was also ready to give “legal force” to the commitments made in a letter in January from Donald Tusk, the president of the European council, and his commission counterpart, Jean-Claude Juncker.

UK sources said there was nothing in the offer that could change MPs’ minds. “It will not be enough to persuade Geoffrey Cox to revise his legal advice about the indefinite nature of the backstop – there is no reason for optimism,” one official close to the negotiations said.

A senior EU official admitted that Barnier’s response could be seen as a “slap in the face”.

Before making public his offer, Barnier had briefed ambassadors to the 27 in what was said to have been a “gloomy” meeting. “There has been a total breakdown in trust,” one EU diplomat said. “They wanted the UK-wide [backstop] and now they don’t like it any more,” added a second. “It’s a strange situation, but there is nothing that should surprise us in British politics.”

“It is very clear that we are running out of time and that you are dealing with a partner preventing solutions to non-existent problems,” the diplomat added.

Comments by the foreign secretary, Jeremy Hunt, have added to a sense of frustration with Westminster. “We are confronted with absolute populism and that is hard to deal with,” said the diplomat.